If the ages of P and R are added to twice the age of Q, the total becomes 59. If the ages of Q and R are added to thrice the age of P, the total becomes 68. And if the age of P is added to thrice the age of Q and thrice the age of R, the total becomes 108. What is the age of P?

Options

A. 15 years
B. 19 years
C. 17 years
D. 12 years

Answer

To solve the problem, we need to set up a system of equations based on the information provided about the ages of P, Q, and R. Let's denote the ages as follows:

- Let P = age of P
- Let Q = age of Q
- Let R = age of R

From the problem statement, we can derive the following equations:

1. From the first statement: "If the ages of P and R are added to twice the age of Q, the total becomes 59."
\[ P + R + 2Q = 59 \quad \text{(Equation 1)} \]

2. From the second statement: "If the ages of Q and R are added to thrice the age of P, the total becomes 68."
\[ 3P + Q + R = 68 \quad \text{(Equation 2)} \]

3. From the third statement: "If the age of P is added to thrice the age of Q and thrice the age of R, the total becomes 108."
\[ P + 3Q + 3R = 108 \quad \text{(Equation 3)} \]

Now we have a system of three equations:

1. \( P + R + 2Q = 59 \) (Equation 1)
2. \( 3P + Q + R = 68 \) (Equation 2)
3. \( P + 3Q + 3R = 108 \) (Equation 3)

### Step 1: Solve Equations 1 and 2

We can manipulate these equations to eliminate one variable. Let's start by isolating \( R \) from Equation 1:

\[ R = 59 - P - 2Q \quad \text{(Substituting into Equation 2)} \]

Now substitute \( R \) into Equation 2:

\[ 3P + Q + (59 - P - 2Q) = 68 \]

Simplifying this gives:

\[ 3P + Q + 59 - P - 2Q = 68 \]
\[ 2P - Q + 59 = 68 \]
\[ 2P - Q = 9 \quad \text{(Equation 4)} \]

### Step 2: Solve Equations 1 and 3

Next, we can also substitute \( R \) into Equation 3:

\[ P + 3Q + 3(59 - P - 2Q) = 108 \]

This simplifies to:

\[ P + 3Q + 177 - 3P - 6Q = 108 \]
\[ -2P - 3Q + 177 = 108 \]
\[ -2P - 3Q = -69 \]
\[ 2P + 3Q = 69 \quad \text{(Equation 5)} \]

### Step 3: Solve Equations 4 and 5

Now we have two equations:

1. \( 2P - Q = 9 \) (Equation 4)
2. \( 2P + 3Q = 69 \) (Equation 5)

We can solve these equations simultaneously. From Equation 4, we can express \( Q \) in terms of \( P \):

\[ Q = 2P - 9 \]

Now substitute this into Equation 5:

\[ 2P + 3(2P - 9) = 69 \]
\[ 2P + 6P - 27 = 69 \]
\[ 8P - 27 = 69 \]
\[ 8P = 96 \]
\[ P = 12 \]

### Conclusion

The age of P is \( \boxed{12} \).